拉尔斯·V. 阿尔福斯（Lars V. Ahlfors） 生前是哈佛大学数学教授。他于1924年进入赫尔辛基大学学习，并在1930年于芬兰著名的土尔库大学获得博士学位。期间他还师从著名数学家Nevanlinna共同进行研究工作。1936年荣获菲尔茨奖。第二次世界大战结束后，他辗转到哈佛大学从事教学工作。1953年当选为美国国家科学院院士。他又于1968年和1981年分别荣获Vihuri奖和沃尔夫奖。他的著述很多，除本书外，还著有Riemann Surfaces和Conformal lnvariants等。
章节目录
Preface CHAPTER 1 COMPLEX NUMBERS1 1 The Algebra of Complex Numbers1 1.1 Arithmetic Operations1 1.2 Square Roots3 1.3 Justification4 1.4 Conjugation, Absolute Value6 1.5 Inequalities9 2 The Geometric Representation of Complex Numbers12 2.1 Geometric Addition and Multiplication12 2.2 The Binomial Equation15 2.3 Analytic Geometry17 2.4 The Spherical Representation18 CHAPTER 2 COMPLEX FUNCTIONS21 1 Introduction to the Concept of Aaalytic Function21 1.1 Limits and Continuity22 1.2 Aaalytic Functions24 1.3 Polynomials28 1.4 Rational Functions30 2 Elementary Theory of Power Serices 33 2.1 Sequences33 2.2 Serues35 2.3 Uniform Convergence35 2.4 Power Series38 2.5 Abels Limit Theorem41 3 The Exponential and Trigonometric Functions42 3.1 The Exponential42 3.2 The Trigonometric Functions43 3.3 The Periodicity44 3.4 The Logarithm46 CHAPTER 3 ANALYTIC FUNCTIONS AS MAPPINGS49 1 Elementary Point Set Topology50 1.1 Sets and Elements50 1.2 Metric Spaces51 1.3 Connectedness54 1.4 Connectedness59 1.5 Continuous Functions63 1.6 Topoliogical Spaces 66 2 Conformality 2.1 Arcs and Closed Curves67 2.2 Analytic Function in Regions69 2.3 Conformal Mapping73 2.4 Length and Area75 3 Linear Transformations76 3.1 The Linear Group76 3.2 The Cross Ratio78 3.3 Symmetry80 3.4 Oriented Circles83 3.5 Families of Circles84 4 Elementary Conformal Mappings89 4.1 The Use of Level Curves89 4.2 A Survey of Elementary Mappings93 4.3 Elementary Riemann Surfaces 97 CHAPTER 4 COMPLEX INTEGRATION101 1 Fundamental Theorems101 1.1 Line Integrals101 1.2 Rectifiable Arcs104 1.3 Line Integrals as Functions of Ares105 1.4 Cauchys Theorem for a Recatangle109 1.5 Cauchys Theorem in a Disk112 2 Cauchys Integral Formula114 2.1 The Index of a Point with Respect to a Closed Curve114 2.2 The Integral Formula118 2.3 Higher Dervatives120 3 Local Properties of Aaalytic Functions124 3.1 Removable Singularites. Taylors Theorem124 3.2 Zeros and Poles126 3.3 The Local Mapping130 3.4 The Mazimum Principle133 4 The General Form of Cauchys Theorem137 4.1 Chains and Cycles 137 4.2 Siple Connectivity138 4.3 Homology141 4.4 The General Statement of Cauchys Theorem141 4.5 Proof of Cauchys Theorem142 4.6 Locally Exact Differentials144 4.7 Multiply Connected Regions146 5 The Calculus of Residues148 5.1 The Residue Theorem148 5.2 The Argument Principle152 5.3 Evaluation of Definite Integrals154 6 Harmonic Functions162 6.1 Definition and Basic Properties162 6.2 The Mean-value Property165 6.3 Poissons Formula168 6.4 Schwarzs Theorem 168 6.5 The Reflection Principle172 CHAPTER 5 SERIES AND PRODUCT DEVELOPMENTS175 1 Power Serices Expansions175 1.1 Weierstrasss Theorem175 1.2 The Taylor Series179 1.3 The Laurent Series184 2 Partial Fractions and Factorzation187 2.1 Partial Fractions187 2.2 Infinite Products191 2.3 Canonical Products 193 2.4 The Gamma Function198 2.5 Stirlings Formula201 3 Entire Functions206 3.1 Jensens Formula207 3.2 Hadamards Theorem208 4 The Riemann Zeta Function212 4.1 The Product Development213 4.2 Extension of (s）to the Whole Plane214 4.3 The Functioal Equation216 4.4 The Zeros of the Zeta Functaion218 5 Normal Families 219 5.1 Equicontinuity219 5.2 Normality and Compactness220 5.3 Arzelas Theorem222 5.4 Families of Analytic Functions223 5.5 The Claaical Definition225 CHAPTER 6 CONFORMAL MAPPUNG. DIRICHLETS PROBLEM229 1 The Riemann Mapping Throrem229 1.1 Statement and Proof229 1.2 Boundary Behavior232 1.3 Use of the Reflection Principle233 1.4 Analytic Arcs234 2 Conformal Mapping of Polygons235 2.1 The Behavior at an Angle 235 2.2 The Schwarz-Christoffel Formula236 2.3 Mapping on a Rectangle238 2.4 The Triangle Functions of Schwarz241 3 A Closer Look at Harmonic Functions241 3.1 Functions with the Mean-value Property242 3.2 Harnacks Principle 243 4 The Dirichlet Problem245 4.1 Subharmonic Functions245 4.2 Solution of Dirchlets Problem248 5 Canonical Mappings of Multiply Connected Regions251 5.1 Harmonic Measures252 5.2 Greens Function 257 5.3 Parallel Slit Regions 259 CHAPTER 7 ELLIPTIC FUNCTIONS263 1 Simply Periodic Functions263 1.1 Representation by Exponentials263 1.2 The Fourier Development264 1.3 Functions of Finite Order264 2 Doubly Periodic Functions265 2.1 The Period Module265 2.2 Unimodular Transformations266 2.3 The Canonical Basis268 2.4 General Properties of Elliptic Functions270 3 The Weierstrass Theory272 3.1 Tht Eeierstrass g-function272 3.2 Tht Functions g(z) and s(z)273 3.3 Tht Differential Equation275 3.4 Tht Modular Function l(t)277 3.5 Tht Conformal Mapping by l(t)279 CHAPTER 8 GLOBAL ANALYTIC FUNCTIONS283 1 Analytic Contrnuation283 1.1 The Weierstrass Theory283 1.2 Germs and Sheaves284 1.3 Sections and Riemann Surfaces287 1.4 Analytic Continuations along Arcs289 1.5 Homotopic Curves291 1.6 The Monodromy Theorem295 1.7 Branch Points297 2 Algebraic Functions300 2.1 The Resultant of Two Polynomials300 2.2 Definition and Properties of Algebraic Function301 2.3 Behavior at the Critical Points304 3 Picards Theorm306 3.1 Lacunary Values307 4 Linear Differentail Equations308 4.1 Ordinary Points309 4.2 Regular Singular Points311 4.3 Solutions at Infinity313 4.4 The Hypergemetric Differential Equation315 4.5 Riemanns Point of View318 Index323
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